Simultaneous matchings: Hardness and approximation

Given a bipartite graph G=(X@?@?D,E@?XxD), an X-perfect matching is a matching in G that covers every node in X. In this paper we study the following generalisation of the X-perfect matching problem, which has applications in constraint programming: Given a bipartite graph as above and a collection F@?2^X of k subsets of X, find a subset M@?E of the edges such that for each C@?F, the edge set M@?(CxD) is a C-perfect matching in G (or report that no such set exists). We show that the decision problem is NP-complete and that the corresponding optimisation problem is in APX when k=O(1) and even APX-complete already for k=2. On the positive side, we show that a 2/(k+1)-approximation can be found in poly(k,|X@?D|) time. We show also that such an approximation M can be found in time (k+(k2)2^k^-^2)poly(|X@?D|), with the further restriction that each vertex in D has degree at most 2 in M.